MOMENTUM, HEAT, AND MASS TRANSFER ANALOGY FOR TURBULENT FLOW IN CIRCULAR PIPES G . H. H U G H M A R K Ethyl Gorp., Baton Rouge, La. 70821
A numerical integration of equations for the turbulent flow velocity profile has been used to calculate heat and mass transfer coefficients inside a smooth circular pipe. The Marchello and Toor mixing model is used for thermal and mass diffusivity in the wall region. Assumption of thermal diffusivity equal to momentum diffusivity for the central core gives good agreement with experimental heat transfer data. An eddy frequency determined from the Shaw and Hanratty experimental data is superimposed on the region near the pipe wall to agree with experimental mass transfer data at Schmidt numbers up to 100,000. The model is also applicable to non-Newtonian fluids because the local apparent viscosity can be related to the local shearing stress.
ECENT
experimental data for turbulent flow mass transfer
R in smooth circular pipes at Schmidt numbers of 450 to 100,000 (Harriott and Hamilton, 1965; Kishinevsky et al., 1966) provide a severe test of the correlations for momentum and mass transfer. The data of Harriott and Hamilton show that for a Reynolds number of 10,000, only the Friend and Metzner (1958) correlation gives reasonable agreement with the experimental mass transfer coefficients. Several attempts have been made to use the velocity profile for turbulent flow to obtain temperature or concentration profiles a t uniform flux and thus predict heat or mass transfer coefficients. Martinelli (1947) divided the velocity profile into three parts: 1. The “laminar sublayer” for r + from 0 to 5 in which e = O
2. A buffer layer for r + from 5 to 30 in which B = 6 , 3. The turbulent core for r+ above 30, also for which € =
e,
Lin et al. (1953) recognized turbulence in the “laminar sublayer” and suggested introduction of an eddy of magnitude E / Y = (r+/14.5)a into the laminar layer. Kropholler and Carr (1962) used the eddy concept of Lin et al. with a four-region velocity profile to obtain a correlation for heat and mass transfer. Total diffusivity for momentum, heat, and mass was assumed to be equal. The purpose of this paper is to develop a model with the velocity profile that will fit the wide range of heat and mass transfer data that are now available for turbulent flow.
For the region from ro+/5 to r+ = 33 : u+ = 5.5
+ 2.5 In r +
For the region from r+ = 33 to u+ = 3.5
T+
=
(2)
22:
+ 2.71 In r +
(3)
The region from r+ = 22 to the pipe wall is very important for the systems of high Schmidt numbers. Longwell (1966) suggests an equation of the form reported by Dunn (1951) for this region : u+ = -tanh (0.0792 r+ )
0.0792
(4)
The total conductivity data of Venezian and Sage for air (1961) apply to the region from r + = 15 to rf = 1. If the molecular and turbulent Prandtl numbers are assumed to be equal, Equation 5 will fit these data for the range of r f = 15 to r+ = l and Equation 4 fit at r + = 15. Thus, Equation 4 is applicable from r + = 22 t o r + = 15.
+ 1.028 r+ - 0.0201 (r+ )z f 0.00123 (r+)a - 0.000432 (r+ )4 +
u+ = -0.0107
- 0.00000087 ( r + ) 6 (5) for r+ < 1, so the form of
0.0000351 ( r + ) s
There are no experimental data Equation 4 was used to fit Equation 5 at r+ = 1. The resulting equation for r+ < 1 is: u + = - tanh (0.078 r + )
0.078
Velocity Profile The high Schmidt number mass transfer data are of particular interest because of the large concentration gradients that occur near the pipe wall. Thus an accurate description of the “laminar sublayer” is required. A six-region velocity profile is used in this work. Equations stated by Hinze (1959) are used to describe the profile from the pipe centerline to r+ = 22. The equations for these three regions are: For the region from the center line to ro+/5:
+
where ug = 4 . 2 5 ~ *
Total Diffurivity
Longwell (1966) presents the following equation for the average Nusselt number a t a uniform heat flux :
N N= ~ ugt?/&
(7)
in which
ub
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31
io3
-
s u)
z L
zP 102
-
*--
*”-
A
BERNARDO ~ E I A N EAGLE a FERGUSON
FRIEND a METZNER HARRIOTT 8 HAMILTON v HOFFMAN
0
0
101 I
I
I
I
I
10
I02
to3
10‘
Npr
Figure 1.
Or
1945 1930 1958 1965 1953
IC
NSO
Heat and mass transfer data at N R = ~ 10,000
Thus the local total diffusivity and the local mean velocity are required for solution of Equation 8. The total diffusivity is the sum of the molecular and eddy diffusivities. For momentum : E,
= en
+
v
(9)
Equations 1 through 6 for the velocity profile can be used to obtain the total diffusivity for momentum:
Numerical Integration
A numerical solution with a digital computer was used for Equation 8. Equations 1 through 6 were used to obtain the local values of u from u = u+/u* and the local value of the total diffusivity for momentum from Equation 10. This program was then used to test several models. A model was tried using the estimate of eddy diffusivity in the “laminar sublayer” according to Lin et al. and total diffusivity for heat equal to total diffusivity for momentum. This model results in calculated heat transfer coefficients about 30% less than experimental values at NRe = 10,000 and Npr = 100. Marchello and Toor (1963) have derived a mixing model to represent heat, mass, and momentum transfer near a fluid boundary. This model assumes that, at low turbulence levels, random mixing of fluid elements occurs rather than gross displacement of fluid elements. The expression for the total momentum eddy diffusivity is:
and for the total mass or thermal diffusivity :
The term E is obtained by eliminating p from Equations 11 and 12 and applying Equation 10 for em, where r + < 33. p is a turbulence parameter with the significance that the ratio of the turbulent to molecular diffusivity is a function only of 32
I&EC FUNDAMENTALS
-.-0.01
0. I (‘0
Figure 2.
I.o
- r)/ro
Heat transfer data for air
Seban and Shimazaki (1 951 1 data
P/D. This model predicts that the total thermal or mass diffusivity is represented by the equation : E
= emdD7
at large values of the total momentum diffusivity. This represents the penetration theory with equal contact times for momentum, heat, and mass. Two models can be formulated:
1. Equation 13 can be used for the freely turbulent region assumed to be 7 + > 33 and Equations 11 and 12 for 7 + < 33. 2. The eddy diffusivities for heat, mass, and momentum are equal in the freely turbulent region and Equations 11 and 12 apply near the wall. A transition region is then required between the wall region and the turbulent core because Equation 13 represents E at the outer limit of the wall region and the core equation is E = em. A range of r + from 25 to 33 was arbitrarily selected for this region with E = E,, at 7 + = 33 and e equal to the value obtained from Equations 11 and 12 at r + = 25. A linear fit of the two values of E was used for this transition region. Figure 1 shows calculated values for the two models with experimental heat transfer data. Model 2 shows better agreement with the experimental data. The two models were also tested with the temperature profile data of Seban and Shima-
Table 1.
System
Calculated and Experimental Heat Transfer Coefficients for Model 2
No. of Reference
Runs
Water
Dipprey and Sabersky (1963)
4
Corn sirup Mercury
Friend and Metzner (1958) Isakoff and Drew (1951)
8 9
Nps
1.23 5.9 59 0.017
N RX ~ 70-4
CT, A v . A issolute Deviation
2.4-12 10.3-50 1.2-3.8 3,9-37
4 2 3 12
zaki (1951) for air and of Isakoff and Drew (1951) for mercury. Again Model 2 shows better agreement with the d a t a . Figure 2 shows this result for the air data. Table I presents the average absolute deviation between calculated heat transfer coefficients from Model 2 and experimental data for several systems. Agreement is very good for this range of Prandtl and Reynolds numbers. GoLven and Smith (1967) obtained temperature profile data for water (.XrP, = 5.7) and aqueous ethylene glycol (LXTpr = 14.3) for the Reynolds number range of 10,000 to 50,000. These data indicate that €/ern is about 0.85 for the turbulent core. The ratio of unity used in Model 2 is obviously a better assumption than the ratios of 0.42 and 0.25 which are calculated from Equation 13 for these fluids with .Model 1. Mass Transfer
Figure 1 shows calculated values of the Sherwood number from Models 1 and 2 compared to experimental data. Model 2 shokrs agreement with data for Schmidt numberi less than 2000 and an increasing deviation to the upper limit of the data at a Schmidt number of 100,000. The high Schmidt number data represent systems in which most of the mass transfer resistance is near the pipe wall and Equations 5 and 6 become very important. Lin et al. (1953) and Son and Hanratty (1967) suggest modification of the velocity profile close to the wall to fit the mass transfer data. Hughmark (1968) extended this approach to the Harriott and Hamilton data. However, equations derived to fit the mass transfer data are inconsistent with the Venezian and Sage (1961) heat transfer data for the region close to the wall. Shaw and Hanratty (1964) obtained experimental data for fluctuations in the local rate of mass transfer in the region close to the pipe wall. These are relatively low frequency fluctuations which indicate random eddies that approach the pipe wall. Harriott (1962) has proposed a model for random eddies superimposed on boundary layer flow in which the eddies approach the wall.
where the first term represents the boundary layer resistance and the second term is the resistance corresponding to the penetration theory representation of the eddies. The concept of additive resistances can be applied to represent the over-all resistance from the pipe center line to the pipe wall as the sum of two resistances: 1. The resistance from the pipe center line to a fixed point in the boundary layer which is assumed to be the point at which the random eddies in the boundary layer are effective. This resistance is designated as l/kB. 2. The resistance from the fixed point in the boundary layer to the pipe wall. This resistance is attributed to the sum of the conductance of the boundary layer, kBL, and the conductance corresponding to the eddies, kPT.
B
Figure 3. transfer
Average frequency of local rate of mass Shaw and Hanratty ( 1 964) daia
This results in the equation:
I t is recognized that the eddies are random in frequency and approach to within random distances of the pipe wall, but it can be assumed that a mean frequency and mean distance from the wall can be used as representative values. The penetration theory model would be expected to predict the mass transfer effect of the eddies. Frequencies of eddies from the Shaw and Hanratty work are applicable to this model. These data were determined in the Reynolds number range of 10,400 to 60,100 with a 1-inch pipe. A dimensionless frequency
n+ = n v / t P was suggested. Figure 3 shows the experimental average frequency plotted as a function of U * ~ / Y . T h e penetration theory can then be used to estimate the mass transfer coefficient from this frequency:
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33
The numerical solution of the boundary layer model can be used to calculate kB and kgL for an assumed approach distance from the pipe wall for the eddies. These values with kPT from Equation 16 are then substituted in Equation 15 to obtain the mass transfer coefficient. Trial and error application of Model 2 to the data of Harriott and Hamilton indicates that the division between kB and kBL occurs at r+ = 0.45. This model with Equation 15 is designated as Model 2A. Figure 1 compares calculated and experimental data for a Reynolds number of 10,000. These data represent a frequency range of 0.18 to 1.35 second-’ from Figure 3, and Figure 1 shows reasonable agreement with the experimental mass transfer data. Model 2A also calculates mass transfer coefficients that give good agreement for the Harriott and Hamilton data for the Reynolds number range of 10,000 to 100,000. Mass transfer data are also reported by Linton and Sherwood (1950) for 1.0- and 5.23-cm. diameter pipe, by Meyerink and Friedlander (1962) for 2.42-cm. pipe, and by Kishinevsky et al. (1966) for 1.3-cm. pipe. The Meyerink and Friedlander and Linton and Sherwood data for the 5.23-cm. diameter benzoic and cinnamic acid sections are consistent with the Harriott and Hamilton data for the 1.9-cm. benzoic acid pipe, and the Kishinevsky et al. experimental data show significantly higher mass transfer coefficients. Linton and Sherwood observed cracks in the pipe wall that could have caused the high coefficients. There is no apparent explanation for the high coefficients in the 1.3-cm. pipe. Non-Newtonian Fluid Heat Transfer
Numerical solution of the velocity profile equations has a distinct advantage for nowNewtonian fluids because the local apparent viscosity can be represented as a function of the local shearing stress. Bogue and Metzner (1963) report velocity profile data for the turbulent core region of non-Newtonian power law fluids. The data were obtained for fluids free of significant viscoelastic effects with a range of flow behavior indices between 0.45 and 0.90. The core profiles obtained with smooth tubes were essentially the same as those for Newtonian fluids when normalized with respect to the mean velocity. Thus the model used in this paper can be used for these power law fluids by defining:
The parameter (,+)‘in represents the conventional r+ with an apparent viscosity evaluated a t the wall shear stress. Friend and Metzner (1959) report heat transfer data for similar fluids. If the Newtonian and non-Newtonian velocity profiles are also similar in the wall region, the model for the Newtonian fluids should be applicable to the non-Newtonian heat transfer data when: Equations 1 7 and 18 are used for r+. The apparent viscosity at each radial position in the tube is defined by the equation :
For power law fluids, K and n vary with temperature. The Friend (1957) heat transfer data were obtained with relatively low temperature differences between the wall and bulk of the fluid, so constant values of K and n corresponding to the bulk 34
IhEC FUNDAMENTALS
Table II.
Calculated Experimental Non-Newtonian Heat Transfer Coefficients yGAverage Absolute Deviation Friend Peterson and No. of and Metzner Christiansen Velocity Runs (1959) ( 1966) profile
Carbopol A
B C Attagel D E F
6 10 7
3 7 4
8.2 15.5a 27.85
6.6
6.4
14.5a
15.2
16.6a 25.55
32.3~ 19.5 41.35
43. Oa 21.5 31.9a
42.5. 19.9 37.5a
- a All deviations are positive-i.e., than experimental.
Table 111.
Variable
all calculated coe8cients are higher
Independent Variables for Equation 20
Friend and Metzner (1959)
Peterson and Christiansen ( 1966)
Velocity Projile
58.0 75 .O 61 . O a 1.2 1.18 1.22 1.02 em 1.0 1.02 a Calculated from Equation 20 with calculated heat transfer coe8cient and 6, = 1.22 and em = 7.02.
NP~ dm
fluid temperature were used. Friend and Metzner (1958) suggested an equation for heat transfer with Newtonian fluids :
Constant values of 1.0 and 1.2 were proposed for 0, and , @ , respectively. Friend and Metzner (1959) applied Equation 20 to the Friend non-Newtonian heat transfer data with a Prandtl number evaluated using the viscosity of the fluid at the wall shearing stress. Peterson and Christiansen (1966) modified Equation 20 for pseudoplastic fluids in turbulent flow by defining an effective Prandtl number to compensate for the apparent viscosity in the low shear rate central region of the tube. Correlations for the ratio of the maximum to the mean velocity (4,) and the maximum to the mean temperature difference (0,) are also proposed. The result is better agreement between calculated and experimental values for the Friend data. Table I1 shows the average absolute deviation between calculated and experimental coefficients for the Friend data with the two semiempirical correlations and as obtained from the velocity profile. Heat transfer coefficients calculated from the velocity profile and the apparent viscosity for the Friend data are nearly the same as those calculated from the Friend and Metzner equation. These calculated coefficients differ significantly from the coefficients calculated by the Peterson and Christiansen correlations. Table I11 compares the values of the three independent variables for the three methods for Friend run C-4. This represents a fully developed turbulent flow condition with an intermediate value of the flow behavior index (n = 0.65). This comparison indicates that the effective viscosity is not as high as shown by the Peterson and Christiansen correction, because the velocity profile method represents the apparent viscosity as a function of the local shearing stress. Observed differences between calculated and experimental heat transfer coefficients are then a result of: Differences in the velocity profile in the region between the wall and the turbulent core.
Lack of applicability of the eddy diffusivity for heat-eddy diffusivity for momentum relationship applied to Newtonian fluids to some of the non-Newtonian fluids tested by Friend.
GREEKLETTERS == defined
=
Other Heat and Mass Transfer Conditions
Numerical solution of the velocity profile can also be applied to conditions of large temperature difference for heat transfer and high mass flux for mass transfer. Large temperature differences bem een the pipe wall and the pipe center line can result in significant local physical property differences, particularly viscosity. Equations for the temperature response of the properties and for the temperature profile can be included in the numerical solution to calculate the heat transfer coefficient. LVasan, Sankar, and Randhave (1967) have suggested equations to modify the velocity profile for high mass fluxes. These modifications can be included with the numerical solution when this condition applies. Summary
Equations for the velocity profile and total diffusivity for turbulent flow in a smooth circular pipe have been integrated to obtain calculated heat and mass transfer coefficients. Use of the Marchello and Toor mixing model for the region near the wall and thermal diffusivity equal to momentum diffusivity for the central core give good agreement with experimental heat transfer data. Models using the Marchello and Toor mixing model and an eddy superimposed on boundary layer flow gave good agreement with experimental mass transfer data. Nomenclature
D
=
D
= molecular diffusivity, sq. ft./hr.
f
Fanning friction factor mass velocity, lb.m/hr. sq. ft. = conversion factor, 4.18 X 108 ft. lb.,/lb.t hr.2 = heat transfer coefficient, B.t.u./hr. sq. ft. O F . = mass transfer coefficient, ft./hr. = thermal conductivity, B.t.u./hr. sq. ft. “F./ft. = thermal diffusivity, sq. ft./hr. (Equation 7) = consistency index, lb., hr.n/sq. ft. (Equations 18 and 19) = frequency, l/sec. (Equation 16) = flow behavior index, dimensionless (Equations 17, 18, and 19) = Xusselt number, 2hro/k’ = Prandtl number, v / K = Reynolds number, 2roub/v = Schmidt number, v / D = Sherwood number, 2 k r 0 / 9 = distance from center of pipe, ft. = pipe radius, ft.
G
gC
h k k‘
K K n
n
r ro r+ ro t
+
Ub
uo U U* Ut
thermal or mass diffusivity, sq. ft./hr.
= =
= (ro
= = = = =
- r ) U*/V
rou*/v
contact time, hr. mean velocity, ft./hr. velocity at center of pipe, ft./hr. time average axial velocity at any radial position, ft ./hr. = friction axial velocity, ft./hr.
ubdfT,
= u/u*
by Equation 11
= eddy diffusivity, sq. ft./hr. = total diffusivity for sq. ft./hr.
eddy diffusivity for momentum, sq. ft./hr.
= total diffusivity for momentum, sq. ft./hr.
ratio of maxmium to mean temperature difference, dimensionless = boundary layer film thickness, ft. = apparent viscosity evaluated at wall shear stress, Ib ,/ft. hr. = kinematic viscosity, sq. ft./’hr. = shear stress at tube wall, lb,f/ft. = ratio of maximum to mean velocity, dimensionless =
V T W
dn,
Literature Cited
Bernardo, E., Eian, C. S., Natl. Advisory Comm. Aeronaut. ARR E5F07 (1945). Bogue, D. C., Metzner, A. B., IND.ENG.CHEM.FUNDAMENTALS 2, 143 (1963). Dipprey, D. F., Sabersky, R. H., Intern. J , Heat Mass Transfer 6, 329 (1963). Dunn, L. G., Powell, SV. B., Seifert, H. S., “Heat Transfer Studies Relating to Power Plant Development,“ Royal -4eronautical Society, 3rd .Anglo-American Aeronautical Conference, 1951. Eagle, A., Ferguson, R. M., Proc. R L ~Sod. . London, A127, 540 (1930). Friend, F. S., ?d.Ch.E. thesis, University of Delaware, Newark, Del., 1957. Friend, TV. L., Metzner, A. B., A.1.Ch.E.J. 4, 393 (1958). Friend, P. S.,h,fetzner, A. B., 2nd. Eng. Chem. 51, 879 (1959). Gowen, R. A,Smith, J. TY., Chem. Eng. Sci. 22, 1701 (1967). Harriott, P.: Chem. Eng. Sci. 17, 149 (1962). Harriott, P., Hamilton, R. M., Chem. Eng. Sci. 20, 1073 (1965). Hinze, J. O., “Turbulence,” pp. 526, 536, McGraw-Hill, New York, 1959. Hoffman, H. W,.“1953 Heat Transfer and Fluid .Mechanics Institute,” p. 83, Stanford University Press, Stanford, Calif., 1953.
Hughmark, G. .4., A.Z.Ch.E.J. 14, 352 (1968). Isakoff, S.E., Drew, T. B., Proceedings of General Discussion on Heat Transfer, In3titution of Mechanical Engineers, London, and American Society of Mechanical Engineers, New York, 1951. Kishinevsky, M. K., Denisova, T. B., Parmenov, V. A,, Intern. J . Heat Mass Transfer 9, 1449 (1966). Kropholler, H. W., Carr, A . D., Intern. J . Heat Mars Transfer 5 , 1191 (1962). Lin, 6. S., Moulton, R. SV., Putnam, G. L., Znd. En,c. - Chem. 45, 636 (1953). Linton. W. H.. Sherwood. T. K.. Chem. Ene. Prom. 46.258 (1950). Longwell, P. Ai.,“Mechanics of Fluid Flow,” pp. 325, 356, 357, McGraw-Hill, New York, 1966. Marchello, J. M., Toor, H. L., IND.ENG.CHEM.FUNDAMENTALS 2, 8 (1963). Martinelli, R. C., Trans. A.S.M.E. 69, 947 (1947). Meyerink, E. S. C., Friedlander, S. K., Chem. Eng. Sci. 17, 121 (1962). Peterson, A. LV., Christiansen, E. B., A.Z.CI2.E.J. 12, 221 (1966). Seban, R. A., Shimazaki, T. T., Trans. A.S.M.E. 73, 803 (1951). Shaw,R. V.,Hanratty,T. J., A.1.Ch.E.J. 10,475 (1964). Son, J. S., Hanratty, T. J., A.1.Ch.E.J. 13,689 (1967). Venezian, E., Sage, B. H., A.1.Ch.E.J. 7, 688 (1961). Wasan, D. T., Sankar, N., Randhave, S.S., A.1.Ch.E. Meeting, Salt Lake City, 1967.
- -
RECEIVED for review September 11, 1967 ACCEPTED August 30, 1968
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